Piriform

 

The piriform is a quartic curve, also known as the pear-shaped quartic. It is the locus of a point P satisfying the following definition.
Piriform[4]. Let C be a circle and L a line. Let O be on the circumference of C such that the diameter through O is perpendicular to L. Pick up a point S on the circumference and draw the perpendicular to L through S, intersecting L at A. The intersection of OA and the parallel to L through S is the point P.
Here is a screenshot of the curve drawn with Jeometry.

Let O be the origin (0, 0) and let L be the line x = a. Let S be (p, q), with (p - r)2 + q2 = r2, where r is the radius of C. Then A will be (a, q) and the point P(x, y) will satisfy the equations

(p - r)2 + q2 = r2
x = p
ay = qx

Eliminating p and q (as usual, you can do it by hand, or use the Groebner applet), we get the Cartesian equation

x4 - 2rx3 + a2y2 = 0.

The piriform has a cusp in O and it is tangent to the circumference at (2r, 0).

We can find a rational parametrization of the piriform by letting y = tx2 and solving for x first and y second. Here is the result:

  2r
x = -------------
  1+a2t2
   
  4r2t
y = -------------.
  (1+a2t2)2

We note that the parametrization does not cover the origin.

 

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